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The Actuary The magazine of the Institute & Faculty of Actuaries

On the analogical actuary

‘What? when I say: “Nicole, bring me my slippers, and give me my night-cap’, is that prose?’ ‘Yes, Sir.’ ‘Good heavens! For more than 40 years I have been speaking prose without knowing it.’Molière’s Bourgeois Gentilhomme, Monsieur Jordain, was pleasantly surprised to learn that he had been talking prose all his life. I suspect that many of us have experienced a similar epiphany on finding that even the simplest actuarial formulae we use, including those written on the backs of envelopes and the fronts of napkins, can generally be classed as models. Almost all of what we do, whether in product pricing, pension fund valuation, or claims reserving, is based on the construction and application of models; it is surprising, therefore, to find that so little has been written about the philosophical basis of modelling.How do we derive results using models? The world of prospective logical derivation is based on two forms of reasoning: deductive and inductive. Deductive reasoning is founded on Aristotle’s syllogisms, the familiar ‘if all A are B, and all B are C, then all A are C’; inductive reasoning, ‘invented’ by Bacon nearly 2,000 years later, is essentially the assumed continuation of rules suggested by evidence. However, modelling does not rely much on these methods (leaving aside the inductive basis of much parameter usage). So what does it rely on?Think about what we are doing in the construction of a model, and the interpretation of its results: we derive a model A’ to represent the world A:A’ when projected leads to B’; from B’ we then de-derive some future world B. We believe that the relationship between world and starting model remains constant when we attempt to deduce something about the new world from the results of our model; thus, we argue by analogy (I am not saying that A is analogous to A’, which would be both trivial and incorrect – strictly, an analogy presupposes two relationships, not one relationship between two objects).Surprisingly, although books abound dedicated to deductive and inductive logic, as regards the analogy we find ourselves in an unscholarly vacuum. Study of the concept has remained confined largely to the fields of rhetoric and literature, ever since Aristotle’s brief discussion in his Art of Rhetoric. However, it can be useful to think about analogical reasoning in order to appreciate better what it is we do all day, and the problems that may afflict our work.Consider the basis of our modelling: as discussed above, if A (the world) is associated with A’ (the modelled world), and if A’ projected gives B’, we deem B’ to be associated with world B. The obvious problem (leaving aside errors in the passage from A’ to B’) lies in the assumed constancy of the A:A’, B:B’ relationship. We assume that that relationship does not change; in other words, we do not think that the unmodelled world will change over the projection period in a way that may affect the validity, or applicability, of our model’s results.Another perspective comes from considering the non-isomorphic nature of the world:model relationship. Given the simplification inherent in any model’s representation of the world, many worlds can give the same model (model in the sense of both the data and the set of relationships which operate on the data); conversely, one set of that model’s results may correspond to many different worlds.The quasi-deification of our models brings the danger of reification, the danger that we might labour under what the philosopher and mathematical physicist Alfred Whitehead termed the fallacy of misplaced concreteness: we start to mistake an abstract model for concrete reality.Although we often go to great lengths to ensure the validity of what we have modelled, how often do we add useful qualitative comment on what we have not modelled, and in particular how changes in the unmodelled world may affect how our model’s results should be interpreted? – a particularly gaping lacuna today, with the imminence of developments such as International Accounting Standards and Solvency II… More generally, to what extent are we guilty of underusing the ‘qualitative’ part of our actuarial minds to complement the gaps inherent in any quantitative work based on analogical reasoning? It would be tempting to conclude in the following manner: that the term analogy is, nowadays, found mostly in studies of literary genres such as allegories, folk tales, and fables; that we no doubt think ourselves above dealing in analogies, let alone folk tales and fables; and that it might therefore be a good thing if, to put the consumers of our results in an appropriate frame of mind, we were always to introduce our reports with a suitable tale from the Brothers Grimm or Mr Aesop. On the other hand, although it may seem rather shocking that we base our work on a combination of analogy (model interpretation) and induction (parameter use), we can at least rest assured that this approach places us in respectable company: Bertrand Russell once observed, ‘…-the whole structure of science, as well as the world of common sense, demands the use of induction and analogy if it is to be believed. These forms of inference, therefore, rather than deduction, are those that must be examined if we are to accept the world of science or any world outside our dreams’.